Damping Factor plays one very important role - it can be used to impress customers.
As for NGF - it is a wonderful tool when it is used wisely. It improves pretty much everything (bandwidth, distortions, output impedance etc). It might produce TIM distortions (higher odd harmonics, overshoot in time domain) for faster changing signals, because of increased amps gain caused by late summing of delayed output signal (phase shift). Reducing bandwidth at the input, perhaps to one that amp had without feedback, should prevent TIM. That would require designing a stable wide bandwidth amp with low distortions to start with. The main problem is that designers use cheap parts and poor circuits trying to fix it with deep feedback. For instance, very popular output transitors 2N3055 have very nonlinear h21e (Beta) - a current gain vs current. There are much better choices but they cost more money (2N3055 cost less than $1). NGF is pretty much free.